The algebraic classification, up to isomorphism, of algebras of dimension \(n\)  from a certain variety
defined by a family of polynomial identities is a classical problem in the theory of non-associative algebras.
There are many results related to the algebraic classification of low-dimensional algebras in the varieties of
Jordan, Lie, Leibniz, Zinbiel and many others [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}.
Another interesting direction is the study of one-generated objects.
The description of one-generated, or cyclic, groups is well known: there exists a unique one-generated group of order \(n\) , up to isomorphism.
In the case of algebras, similar results have been obtained in varieties such as associative [12]}, non-commutative Jordan [13]}, Leibniz and Zinbiel [14]}, since it was proved that there is only one \(n\) -dimensional one-generated nilpotent algebra from each of them.
However, this circumstance does not hold for every variety of non-associative algebras. For example, the classifications of 4-dimensional right commutative [15]}, assosymmetric [16]}, bicommutative [17]}, commutative [18]} and terminal [19]} nilpotent algebras show that there exist several one-generated algebras of dimension 4 from these varieties.
Recently,
one-generated nilpotent Novikov and assosymmetric algebras in dimensions 5 and \(6,\)  and one-generated nilpotent terminal algebras in dimension 5 were classified in [20]}, [21]}, [4]}.
In the present paper, we give the algebraic classification of
5- and 6-dimensional complex one-generated nilpotent bicommutative algebras.
